It's one-third, or ~33%.

if we make a three by three matrix, we'll see why:

we'll call the management trainees t1, t2 and t3.
and recall that the quarters are represent by c1, c2, c3.

so here is all the possible combinations:
[t1c1][t1c2][t1c3]
times
[t2c1][t2c2][t2c3]
times
[t3c1][t3c2][t3c3]

so that's 3x3x3 = 27 possible combinations

we can see that anytime all three trainees apply for the same job:

t1cx
t2cx
t3cx

happens three possible times.

3/27 = 1/9 =~ 11.11%